UID:
almafu_9959238161202883
Format:
1 online resource (xiv, 360 pages) :
,
digital, PDF file(s).
ISBN:
1-316-08682-8
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1-107-09073-3
,
1-107-08764-3
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1-107-09988-9
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1-107-09385-6
,
1-139-17188-7
Series Statement:
London Mathematical Society student texts ;
Content:
The p-adic numbers, the earliest of local fields, were introduced by Hensel some 70 years ago as a natural tool in algebra number theory. Today the use of this and other local fields pervades much of mathematics, yet these simple and natural concepts, which often provide remarkably easy solutions to complex problems, are not as familiar as they should be. This book, based on postgraduate lectures at Cambridge, is meant to rectify this situation by providing a fairly elementary and self-contained introduction to local fields. After a general introduction, attention centres on the p-adic numbers and their use in number theory. There follow chapters on algebraic number theory, diophantine equations and on the analysis of a p-adic variable. This book will appeal to undergraduates, and even amateurs, interested in number theory, as well as to graduate students.
Note:
Includes index.
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Reprinted 1988, 1995.
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Cover; Series Page; Title; Copyright; PREFACE; CONTENTS; LEITFADEN; NOTATIONAL CONVENTIONS; CHAPTER ONE: INTRODUCTION; 1 VALUATIONS; 2 REMARKS; 3 AN APPLICATION; CHAPTER TWO: GENERAL PROPERTIES; 1 DEFINITIONS AND BASICS; 2 VALUATIONS ON THE RATIONALS; 3 INDEPENDENCE OF VALUATIONS; 4 COMPLETENESS; 5 FORMAL SERIES AND A THEOREM OF EISENSTEIN; CHAPTER THREE: ARCHIMEDEAN VALUATIONS; 1 INTRODUCTION; 2 SOME LEMMAS; 3 COMPLETION OF PROOF; CHAPTER FOUR: NON-ARCHIMEDEAN VALUATIONS. SIMPLE PROPERTIES; 1 DEFINITIONS AND BASICS; 2 AN APPLICATION TO FINITE GROUPS OF RATIONAL MATRICES; 3 HENSEL'S LEMMA
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4 ELEMENTARY ANALYSIS5 A USEFUL EXPANSION; 6 AN APPLICATION TO RECURRENT SEQUENCES; CHAPTER FIVE: EMBEDDING THEOREM; 1 INTRODUCTION; 2 THREE LEMMAS; 3 PROOF OF THEOREM; 4 APPLICATION. A THEOREM OF SELBERG; 5 APPLICATION. THE THEOREM OF MAHLER AND LECH; CHAPTER SIX: TRANSCENDENTAL EXTENSIONS. FACTORIZATION; 1 INTRODUCTION; 2 GAUSS' LEMMA AND EISENSTEIN IRREDUCIBILITY; 3 NEWTON POLYGON; 4 FACTORIZATION OF PURE POLYNOMIALS; 5 WEIERSTRASS PREPARATION THEOREM; CHAPTER SEVEN: ALGEBRAIC EXTENSIONS (COMPLETE FIELDS); 1 INTRODUCTION; 2 UNIQUENESS; 3 EXISTENCE; 4 RESIDUE CLASS FIELDS; 5 RAMIFICATION
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6 DISCRIMINANTS7 COMPLETELY RAMIFIED EXTENSIONS; 8 ACTION OF GALOIS; CHAPTER EIGHT: P-ADIC FIELDS; 1 INTRODUCTION; 2 UNRAMIFIED EXTENSIONS; 3 NON-COMPLETENESS OF Qp; 4 ""KRONECKER-WEBER"" THEOREM; CHAPTER NINE: ALGEBRAIC EXTENSIONS (INCOMPLETE FIELDS); 1 INTRODUCTION; 2 PROOF OF THEOREM AND COROLLARIES; 3 INTEGERS AND DISCRIMINANTS; 4 APPLICATION TO CYCLOTOMIC FIELDS; 5 ACTION OF GALOIS; 6 APPLICATION. QUADRATIC RECIPROCITY; CHAPTER TEN: ALGEBRAIC NUMBER FIELDS; 1 INTRODUCTION; 2 PRODUCT FORMULA; 3 ALGEBRAIC INTEGERS; 4 STRONG APPROXIMATION THEOREM; 5 DIVISORS. RELATION TO IDEAL THEORY
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6 EXISTENCE THEOREMS7 FINITENESS OF THE CLASS NUMBER; 8 THE UNIT GROUP; 9 APPLICATION TO DIOPHANTINE EQUATIONS. RATIONAL SOLUTIONS; 10 APPLICATION TO DIOPHANTINE EQUATIONS. INTEGRAL SOLUTIONS; 11 THE DISCRIMINANT; 12 THE KRONECKER-WEBER THEOREM; 13 STATISTICS OF PRIME DECOMPOSITION; CHAPTER ELEVEN: DIOPHANTINE EQUATIONS; I INTRODUCTION; 2 HASSE PRINCIPLE FOR TERNARY QUADRATICS; 3 CURVES OF GENUS 1. GENERALITIES; 4 CURVES OF GENUS 1. A SPECIAL CASE; CHAPTER TWELVE: ADVANCED ANALYSIS; 1 INTRODUCTION; 2 ELEMENTARY FUNCTIONS; 3 ANALYTIC CONTINUATION; 4 MEASURE ON Zp; 5 THE ZETA FUNCTION
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6 L-FUNCTIONS7 MAHLER'S EXPANSION; CHAPTER THIRTEEN: A THEOREM OF BOREL AND DWORK; 1 INTRODUCTION; 2 SOME LEMMAS; 3 PROOF; APPENDIX A: RESULTANTS AND DISCRIMINANTS; APPENDIX B: NORMS, TRACES AND CHARACTERISTIC POLYNOMIALS; APPENDIX C: MINKOWSKI'S CONVEX BODY THEOREM; APPENDIX D: SOLUTION OF EQUATIONS IN FINITE FIELDS; APPENDIX E: ZETA AND L-FUNCTIONS AT NEGATIVE INTEGERS; APPENDIX F: CALCULATION OF EXPONENTIALS; REFERENCES; INDEX
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English
Additional Edition:
ISBN 0-521-31525-5
Additional Edition:
ISBN 0-521-30484-9
Language:
English
